Other Pergamon Titles of Interest BARTENEV, G. M. and ZUYEV, Yu. S. Strength and Failure of Visco-elastic Materials BELYAYEV, Ν. M. Problems in Strength of Materials BRITVEC, S. J. The Stability of Elastic Systems CALLADINE, C. R. Engineering Plasticity CHEUNG, Y. K. Finite Strip Method in Structural Analysis CHRISTIAN, J. W. The Theory of Transformations in Metals and Alloys, 2nd edition DUGDALE, D. S. Elements of Elasticity HENDRY, A. W. Elements of Experimental Stress Analysis (SI edition) HERRMANN, G. and PERRONE, Ν. Dynamic Response of Structures HULL, D. Introduction to Dislocations, 2nd edition ILYUSHIN, A. A. and LENSKII, V. S. Strength of Materials JAEGER, L. G. Cartesian Tensors in Engineering Science JAEGER, L. G. Elementary Theory of Elastic Plates KOVACS, I. and ZSOLDOS, L. Dislocations and Plastic Deformation SAADA, A. S. Elasticity: Theory and Applications WASHIZU, K. Variational Methods in Elasticity and Plasticity, 2nd edition STRESS WAVES IN NON-ELASTIC SOLIDS by W. K. NOWACKI Translated by ZBIGNIEW OLESIAK P E R G A M ON P R E SS OXFORD · NEW YORK · TORONTO · SYDNEY PARIS · FRANKFURT U.K. Pergamon Press Ltd., Headington Hill Hall, Oxford 0X3 OBW, England U. S. A. Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523, U.S.A. CANADA Pergamon of Canada Ltd., 75 The East Mall, Toronto, Ontario, Canada AUSTRALIA Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia FRANCE Pergamon Press SARL, 24 rue des Ecoles, 75240 Paris, Cedex 05, France FEDERAL REPUBLIC Pergamon Press GmbH, 6242 Kronberg Taunus, OF GERMANY Pferdstrasse 1, Federal Republic of Germany Copyright © 1978 Pergamon Press Ltd. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers First English Edition 1978 Library of Congress Cataloging in Publication Data Nowacki, Wojciech Krzysztof. Stress waves in non-elastic solids. Bibliography: p. 1. Elastic solids. 2. Stress waves. 3. Visco- plasticity. I. Title. QC191.N68 1977 53Γ.33 76-52418 ISBN 0-08-021294-8 This is a translation of the Polish book Zagadnienia Falowe w Teorii Plastycznosci published by Paiistwowe Wydawnictwo Naukowe, Warsaw. Printed in Great Britain by William Clowes & Sons, Limited London, Beccles and Colchester CONTENTS vii Bibliography 237 243 Name Index 245 Subject Index PUBLISHER'S NOTE In order to make this volume available as economically and as rapidly as possible much of the mathematics has been reproduced directly from the original Polish edition. Due to technical problems, however, it was not possible to match the typography of the new updated material with the original, and certain characters within the new text matter will appear different from those in the reproduced matter. The variations comprise ν which appears as ν in the original, e which appears as ε in the original, g which appears as g in the original; no distinction is intended and it is hoped that they will in no way distract the reader. PREFACE The dynamics of inelastic continua is nowadays a domain of intense study. Society, aiming at its own welfare and safety but aware of limited natural resources, requires reliable answers regarding the mechanical behaviour of solids and the performance of structures under transient agencies. Impacts, blasts, collisions, thermal shocks, and irradiations constitute current hazards of modern technology. Assessments of damage and failure due to the propagation of stress waves, evaluations of permanent deformation and energy absorption under impact, methods of estimation of pressure transmitted through protective structures, calculations of structural reliability under transient random charges, are of importance for the development of technology and for a safe utilization of its products. The last decade is associated with significant advances in the mechanical sciences, in particular as regards the inelastic behaviour of materials and structures. Mathematical models of complex material response were formulated within the framework of non-linear continuum mechanics and justified in often sophisticated experiments. Mechanisms of inelastic deformation were explored to a large extent and couplings between mechanical and other fields were studied. Methods of solving boundary value problems for the differential equations governing the dynamics of inelastic solids were developed and various problems of a direct technological interest regarding structural strength and the propagation of waves in elastic—plastic and rate sensitive materials were solved. Advanced specialized monographs are thus needed to facilitate an access to the accumulated knowledge and to its utilization in engineering service to society. The present book largely contributes towards meeting such a demand in the domain of continuum dynamics. Dr. Nowacki has succeeded in producing both an original and a comprehensive presentation of principles, methods, and solutions regarding the propagation of waves in elastic—plastic and viscoplastic solids. This monograph is comprehensive since it exposes wave propagation problems for the considered types of material response starting with discussions of constitutive relations, justifies the hypotheses introduced in specialized theories and the simplifications made in the analysis of particular problems, presents both analytical and numerical methods of solving problems, and gives a large number of solutions to specific problems of wave propagation in inelastic solids. The book is original both in its outline and in its contents. It includes a number of contributions Dr. Nowacki has made to the study of wave propagation in elastic—plastic and viscoplastic continua. There is a competent exposition of the mathematical questions relating to hyperbolic differential equations, a presentation of analytical and numerical techniques resulting in effective solutions of problems concerning plane, cylindrical and spherical stress waves and thermal stress waves, and discussions of two-dimensional waves, the whole being supplemented by solved problems. All this makes the book useful both in study and in design. ix X PREFACE By its contents and its competent and lucid presentation of problems of inelastic waves the book constitutes a significant and useful contribution to the domain of continuum dynamics. It has no equivalent counterpart in the existing literature and it will doubtless be appreciated by both specialists and students because of its originality and pertinence. Warsaw, 13 April, 1976 A. SAWCZUK INTRODUCTION This book is devoted to wave problems in the theory of plasticity. Such problems arise in cases in which intensive dynamic loads acting on the elements of a structure are big enough to produce plastic deformation of the elements. To date a considerable number of contributions and monographs have been published in this field. The first papers referring to these problems appeared in the forties and the theory rapidly developed in the sixties. Many problems, particularly the one-dimensional ones, have been investigated fully. A few monographs exist presenting the theory of impact in continuous and discrete systems in a general way. Here we should quote, first of all, the monograph by Kolsky [71], published in 1953, next the book by Goldsmith [46] which appeared in 1960, and those by Rakhmatulin and Demianov [125] and Cristescu [34], published in 1961 and 1967 respectively. In these books the reader can find the general theory of impact in continuous plastic bodies. Since the publication of these books a number of problems have been solved which are significant from the standpoint of engineering practice. Numerous papers on the dynamics of inelastic structures have appeared. In connection with the rapid development of computational techniques a great number of methods have been devised for the numerical integration of quasi-linear and semi-linear partial differential equations. These investigations have been accompanied by the development of the pertinent analytical methods of solving initial and boundary value problems for inelastic bodies. They mainly refer to the problems of stress wave propagation in the cases of complicated stress states, the waves generated by multi-parameter loading, three-dimensional waves, and, finally, to thermal stress waves. This book is confined to those wave problems which are such that they can be described by a system of hyperbolic partial differential equations of the first order, which are quasi-linear or semi-linear. A survey of the literature is given in the field of stress wave propagation in elastic/ viscoplastic media and in particular the papers of Polish authors are discussed. The book is arranged in the following way. The first chapter has been devoted to the fundamental equations of the dynamics of inelastic media. The dynamical properties of materials (metals and soils) are discussed offering an account of the most representative theories of plasticity and viscoplasticity. Chapter II considers the basic definitions of discontinuity surfaces and the conditions which have to be satisfied across these surfaces. Simultaneously, certain mathematical fundamentals have been given, referring to systems of differential equations, quasi-linear and semi-linear, of the first order. Also initial and boundary value problems for hyperbolic equations have been formulated. The remaining chapters have been devoted to methods of solving stress wave propagation problems, namely, one-dimensional plane waves, spherical and cylindrical waves, longitudinal—transverse waves, waves in beams and plates, and plane two-dimensional stress waves. Wave propagation problems for elastic-plastic xi xii INTRODUCTION and elastic/viscoplastic media have been treated in detail, as well as the most important problem of shock waves in metals and soils. The last chapter deals with thermal wave propagation problems, relating to waves generated by thermal shocks applied to the boundary of the media under consideration. Wave propagation problems in strings and membranes are not discussed in this book. The reader is referred for an exhaustive treatment to the monographs by Rakhmatulin and Demianov [125] and by Cristescu [34]. Since the time these books appeared the subject has not been changed except for some insignificant modifications. The theory of impact in inelastic solids, in which wave phenomena are not taken into account, have also been neglected in this book. These problems have been discussed in a number of monographs, for example in [118] (chapter V) and in [47] (chapter XV, where the theory of impact in discrete mechanical systems is discussed). I am much indebted and thankful to my colleague Dr. Bogdan Raniecki from the Institute of Fundamental Technical Research of the Polish Academy of Sciences for his valuable suggestions and numerous discussions during the preparation of the book. CHAPTER I FUNDAMENTAL EQUATIONS OF THE DYNAMICS OF INELASTIC MEDIA 1. Dynamical properties of materials The dynamical properties of materials will be briefly presented based on the results of experiments. The reader can find more details of the experiments in the monographs by Rakhmatulin and Demianov [125], Kolsky [71 ], Cristescu [34], and Perzyna [114] and [118]. The investigation of the behaviour of materials under dynamic conditions has taken place in several different directions. The determination of the dynamic characteristic (constitutive relationship) constitutes the primary direction of the investigations. Also, much research has been done in order to find the plastic strain distribution along the specimens tested and to investigate the influence of transverse motion during the propagation of longitudinal waves in a test piece which produces the effect of wave dispersion. The object of numerous experimental investigations has been to determine the influence of temperature and that of radiation on the behaviour of metals under dynamic conditions. In the case of static processes, the mechanical properties of material are described by the stress-strain diagram. In the case of the dynamic loadings, however, the stress—strain relation is influenced by the strain rate. The influence of a number of other phenomena is also observed, e.g. temperature, radiation, etc. The fundamental properties of metals and soils will be presented in this section. 1.1. DYNAMICAL PROPERTIES OF METALS The experimental investigations of Clark and Duwez [25], Manjoine [84], Häuser, Simmons and Dorn [51], Campbell and Ferguson [22], Marsh and Campbell [88], Lindholm [76] - [78], and of many other authors reveal that much higher stress is necessary to reach the yield limit in an impact loading compared with that for a slow one. In the case of a number of practically important materials (e.g. high-carbon steels) a relationship explicitly independent of the strain rate [125] σ = a(e), obtained during dynamic loading of a specimen, can be used to characterize the dynamic behaviour of the materials. This relationship essentially differs from the static characteristic. The dynamic and static characteristics for a specimen made of mild steel are presented in Fig. 1. Curve 1 denotes the static case, while 2 indicates the dynamic one. Curve 2 has been drawn using the method due to Rakhmatulin [125], based on the measurement of plastic deformation of the initial cross-section of the specimen under the effect of the impact as a function of the impact speed. The yield limit of the material increases for dynamic loading of the specimen. The character of the curve is determined by the type of material. It has been 1 2 STRESS WAVES IN NON-ELASTIC SOLIDS found in many experimental investigations that those metals with a distinct yield limit are particularly sensitive to strain rate. A good example of this behaviour of exhibiting high sensitivity to strain rate is that of mild steel and pure iron. The influence of strain rate on the change of yield limit for mild steel was the subject of investigations in [25], [84], [76] - [78], [88], [24], and in other papers. For example, Clark and Duwez [25] found that for mild steel 6.0\- 200 ? εχΙ0 *0'] Fig. 1 Fig. 2 (0.22% carbon) the yield limit increases with the increase of strain rate (Fig. 2) from about 2.71 X 108 Pa (the static yield limit) to a value of about 5.76 Χ 10 8 Pa for a strain rate of e « 200 sA . The yield limit increases up to the moment when it joins the curve representing the change of the conventional strength limit (dashed line in Fig. 2). The conventional strength limit increases in the range of strain rates from 0 to « 200 s" 1. The results of numerous experiments have shown that the yield limit for mild steel in dynamic loading situations can reach a value which is 2—3 times greater than that for static loading processes. It has also been noted that in a dynamic loading process the strain-hardening effect decreases in comparison with that for static loading. Most of the investigations connected with the determination of dynamic material characteristics were performed under conditions of uniaxial tension or compression or for pure shear. Only a few experiments have been performed so far investigating the material characteristics in complex loading conditions. The investigations of lindholm [76] — [78] belong to the pioneer ones in this field. The results of these experiments refer to aluminium and steel specimens under uniaxial tension and shear. The second stress tensor invariant against the second strain tensor invariant is shown in Fig. 3 for various strain rates [76] —[78]. The experiments were performed for a range of strain rates from 10 3 s"1 to A Χ 10"3 s"1. It is clear from the diagram that for higher strain rates the greater values of the stress intensity \fTi correspond to a definite value of the strain intensity y/T^= const. Lindholm [77] has performed identical experiments for steel specimens, obtaining similar results. There already exists an extensive literature concerning the influence of strain rate as well as of temperature on the stress—strain relations. Each of these effects is considered separately. It has been found experimentally that at low temperatures the lower yield limit of metals does not depend on the strain rate, while at elevated temperatures even very small changes of strain rate cause a significant decrease or increase in the yield limit [22], [24].